Physics-aware automatic spatial planning for subtractive and hybrid manufacturing

ABSTRACT

A method includes receiving a representation of a near-net shape including a 3D part and a support volume. The method also includes calculating a measure of inaccessibility of the support volume by at least one subtractive tool assembly. The method also includes calculating a measure of change in a physical quantity of interest with respect to a change in the near-net shape. The method also includes constructing a physics-aware inaccessibility measure based at least partially upon the measure of inaccessibility, the measure of change, or both. The method also includes creating a plan to remove at least a portion of the support volume using the at least one subtractive tool assembly based at least partially upon the physics-aware inaccessibility measure.

TECHNICAL FIELD

The present teachings relate generally to three-dimensional (3D)printing and, more particularly, to automated design generation foradditive manufacturing with an accessible support volume.

BACKGROUND

Additive manufacturing (AM) technologies are capable of fabricatinggeometrically complex parts by adding material layer-by-layer. Thegrowing interest in AM, specifically metal AM, stems from its ability toleverage geometric complexity to design high-performance, light-weightdesigns for applications such as aerospace, automotive, medical, etc.However, in most metal AM technologies, such as powder-bed fusion, asacrificial support volume is used in “overhanging” regions to dissipateexcessive heat and ensure successful build of a near-net shape. As usedherein, “near-net shape” (NNS) refers to the combination of the desired3D part and the support volume, both of which are generated by the 3Dprinter during the AM process.

After printing, the support volume is removed using subtractivemanufacturing (SM) such as milling or machining. Thus, the supportvolume needs to be accessible by the available machining tools andfixtures for removal. SM techniques, such as multi-axis machining, havebeen widely used for manufacturing high-quality reproducible partsacross multiple industries (e.g., aerospace and automotive industries).In SM, a user begins with a raw stock of material (e.g., the NNS), andportions thereof (e.g., the support volume) are carved away until thedesired 3D part emerges.

Therefore, what is needed is an improved system and method for designgeneration for additive manufacturing with an accessible support volume.

SUMMARY

The following presents a simplified summary in order to provide a basicunderstanding of some aspects of one or more embodiments of the presentteachings. This summary is not an extensive overview, nor is it intendedto identify key or critical elements of the present teachings, nor todelineate the scope of the disclosure. Rather, its primary purpose ismerely to present one or more concepts in simplified form as a preludeto the detailed description presented later.

A method is disclosed. The method includes receiving a representation ofan initial design domain. The method also includes iterativelygenerating intermediate part designs by redistributing material withinthe initial design domain. The intermediate part designs each include a3D part and a support volume. The method also includes calculating ameasure of inaccessibility of the support volume of each intermediatepart design by at least one subtractive tool assembly. At least one ofthe intermediate part designs is generated based at least partially uponthe measure of inaccessibility of a previous one of the intermediatepart designs.

A method for generating a design of a 3D part is also disclosed. Themethod includes providing a computer with a representation of an initialdesign domain, a build orientation for building the initial designdomain using an additive manufacturing process, and at least onesubtractive tool assembly with a number of degrees of freedom for asubtractive manufacturing process. The method also includes iterativelygenerating intermediate part designs by redistributing material withinthe initial design domain using the computer. The intermediate partdesigns each include the 3D part and a support volume. The method alsoincludes calculating a measure of inaccessibility of the support volumeof each intermediate part design by the at least one subtractive toolassembly using the computer. A subsequent one of the intermediate partdesigns is generated based at least partially upon the measure ofinaccessibility of a previous one of the intermediate part designs.

A method for generating a design of a 3D part that is to be manufacturedby an additive manufacturing process followed by a subtractivemanufacturing process is also disclosed. The method includes providing acomputer with a representation of an initial design domain, a buildorientation for building the initial design domain using the additivemanufacturing process, and at least one subtractive tool assembly with anumber of degrees of freedom for the subtractive manufacturing process.The method also includes generating an intermediate part design withinthe initial design domain using the computer. The intermediate partdesign includes the 3D part and a support volume. The method alsoincludes calculating a measure of inaccessibility of the support volumeof the intermediate part design by the at least one subtractive toolassembly using the computer. The method also includes generating asubsequent intermediate part design by redistributing material withinthe initial design domain using the computer. The subsequentintermediate part design is generated based at least partially upon themeasure of inaccessibility.

A method is also disclosed. The method includes receiving arepresentation of a near-net shape including a 3D part and a supportvolume. The method also includes calculating a measure ofinaccessibility of the support volume by at least one subtractive toolassembly. The method also includes calculating a measure of change in aphysical quantity of interest with respect to a change in the near-netshape. The method also includes constructing a physics-awareinaccessibility measure based at least partially upon the measure ofinaccessibility, the measure of change, or both. The method alsoincludes creating a plan to remove at least a portion of the supportvolume using the at least one subtractive tool assembly based at leastpartially upon the physics-aware inaccessibility measure.

A method of planning for removal of a support volume in hybridmanufacturing is also disclosed. The method includes providing acomputer with a representation of a 3D part, a near-net shape includingthe 3D part and the support volume, and at least one subtractive toolassembly with a number of degrees of freedom. The method also includescalculating a measure of inaccessibility of the support volume by the atleast one subtractive tool assembly using the computer. The method alsoincludes calculating a measure of change in a physical quantity ofinterest with respect to a change in the near-net shape using thecomputer. The method also includes constructing a physics-awareinaccessibility measure by combining the measure of inaccessibility andthe measure of change using the computer. The physics-awareinaccessibility measure indicates a removability of a region of thesupport volume from the near-net shape. The method also includescreating a plan to remove the region of the support volume with the atleast one subtractive tool assembly using the computer. The plan isbased at least partially upon the physics-aware inaccessibility measure.

A method of planning for removal of a support volume in hybridmanufacturing where the support volume is added during an additivemanufacturing process and subsequently removed through a subtractivemanufacturing process is also disclosed. The method includes providing acomputer with a representation of a 3D part, a near-net shape includingthe 3D part and the support volume, and at least one subtractive toolassembly with a number of degrees of freedom. The method also includescalculating a measure of inaccessibility of the support volume by the atleast one subtractive tool assembly using the computer. The method alsoincludes calculating a measure of change in a physical quantity ofinterest with respect to a hypothetical change in the near-net shape.The hypothetical change is less than a predetermined size. The methodalso includes constructing a physics-aware inaccessibility measure bycombining the measure of inaccessibility and the measure of change usingthe computer. The physics-aware inaccessibility measure indicates aremovability of a region of the support volume from the near-net shape.The method also includes creating a plan to remove the region of thesupport volume with the at least one subtractive tool assembly using thecomputer. The plan is based at least partially upon the physics-awareinaccessibility measure.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of this specification, illustrate embodiments of the presentteachings and together with the description, serve to explain theprinciples of the disclosure. In the figures:

FIG. 1 depicts a cross-sectional view of a 3D printing system, accordingto an embodiment.

FIG. 2 depicts a side view of an initial design and boundary condition,according to an embodiment.

FIG. 3 depicts a side view of the initial design and boundary conditionin a different build orientation (e.g., 45°), according to anembodiment.

FIG. 4 depicts a side view of a near-net shape (NNS) including a 3D partand a support volume, according to an embodiment.

FIG. 5 depicts a schematic view of a subtractive tool assembly forremoving the support volume from the NNS to yield the 3D part, accordingto an embodiment.

FIG. 6 depicts an image of an IMF over the NNS in FIG. 4 , according toan embodiment.

FIG. 7 depicts an image of the IMF over the inaccessible support volumein FIG. 4 , according to an embodiment.

FIG. 8 depicts another side view of the NNS, according to an embodiment.

FIG. 9 depicts an image of the IMF over the NNS in FIG. 8 , according toan embodiment.

FIG. 10 depicts a flowchart of a method for generating a design of theNNS, according to an embodiment.

FIG. 11 depicts perspective view of a portion of the 3D printing systemin FIG. 1 , according to an embodiment.

FIG. 12 depicts another example of the NNS on a build platform,according to an embodiment.

FIGS. 13A-13E depict different fields that are computed for a firstsubtractive tool assembly and its first orientation, according to anembodiment.

FIGS. 14A-14E depict progressive removal of the support volume from theNSS to yield the 3D part, according to an embodiment.

FIG. 15 depicts a flowchart of a method for planning for removal of thesupport volume from the NSS, according to an embodiment.

DETAILED DESCRIPTION

Reference will now be made in detail to exemplary embodiments of thepresent teachings, examples of which are illustrated in the accompanyingdrawings. Wherever possible, the same reference numbers will be usedthroughout the drawings to refer to the same, similar, or like parts.

FIG. 1 depicts a schematic cross-sectional view of a 3D printing system100, according to an embodiment. The 3D printing system 100 may includean ejector (also referred to as a pump chamber) 110. The ejector 110 maydefine an inner volume that is configured to receive a printing material120. The printing material 120 may be or include a metal, a polymer, orthe like. For example, the printing material 120 may be or includealuminum (e.g., a spool of aluminum wire). In another embodiment, theprinting material 120 may be or include copper.

The 3D printing system 100 may also include one or more heating elements130. The heating elements 130 are configured to melt the printingmaterial 120 within the inner volume of the ejector 110, therebyconverting the printing material 120 from a solid material to a liquidmaterial (e.g., liquid metal) 122 within the inner volume of the ejector110.

The 3D printing system 100 may also include a power source 132 and oneor more metallic coils 134. The metallic coils 134 are wrapped at leastpartially around the ejector 110 and/or the heating elements 130. Thepower source 132 may be coupled to the coils 134 and configured toprovide power thereto. In one embodiment, the power source 132 may beconfigured to provide a step function direct current (DC) voltageprofile (e.g., voltage pulses) to the coils 134, which may create anincreasing magnetic field. The increasing magnetic field may cause anelectromotive force within the ejector 110, that in turn causes aninduced electrical current in the liquid metal 122. The magnetic fieldand the induced electrical current in the liquid metal 122 may create aradially inward force on the liquid metal 122, known as a Lorentz force.The Lorentz force creates a pressure at an inlet of a nozzle 114 of theejector 110. The pressure causes the liquid metal 122 to be jettedthrough the nozzle 114 in the form of one or more drops 124.

The 3D printing system 100 may also include a substrate (also referredto as a build plate) 140 that is positioned below the nozzle 114. Thedrops 124 that are jetted through the nozzle 114 may land on thesubstrate 140 and cool and solidify to produce a near-net shape (NNS)126, which may also be referred to as a 3D object. As described ingreater detail below, the NNS 126 may include a desired 3D part (alsoreferred to as the net shape) and a support volume.

The substrate 140 may include a heater 142 therein that is configured toincrease the temperate of the substrate 140. The 3D printer 100 may alsoinclude a substrate control motor 144 that is configured to move thesubstrate 140 as the drops 124 are being jetted (i.e., during theprinting process) to cause the NNS 126 to have the desired shape andsize. The substrate control motor 144 may be configured to move thesubstrate 140 with up to six degrees of freedom (e.g., threetranslations and three rotations). In another embodiment, the ejector110 and/or the nozzle 114 may be also or instead be configured to movewith up to six degrees of freedom.

The 3D printing system 100 may also include one or more subtractive toolassemblies (one is shown 160). The subtractive tool assembly 160 mayinclude one or more holders (one is shown: 162) and one or more cutters(one is shown: 164). The holder 162 may be configured to mount and/orguide the cutter 164 to regions of interest on the NNS 126. The cutter164 may be configured to cut (e.g., mill, machine, etc.) the supportvolume away from the NNS 126 to yield the 3D part. The 3D printingsystem 100 may also include one or more fixturing devices (one is shown:166) that hold the NNS 126 as material is added and/or removedtherefrom. For example, the fixturing device 166 may be configured tohold the NNS 126, after the AM process, as material (e.g., the supportvolume) is/are removed therefrom by the subtractive tool assembly 160.

In one embodiment, the 3D printing system 100 may also include anenclosure 170. The enclosure 170 may be positioned at least partiallyaround the ejector 110, the nozzle 114, the drops 124, the NNS 126, theheating elements 130, the coils 134, the substrate 140, the subtractivetool assembly 160, or a combination thereof. In one embodiment, theenclosure 170 may be hermetically sealed. In another embodiment, theenclosure 170 may not be hermetically sealed. In other words, theenclosure 170 may have one or more openings that may allow gas to flowtherethrough. For example, the gas may flow out of the enclosure 170through the openings.

In one embodiment, the 3D printing system 100 may also include one ormore gas sources (one is shown: 180). The gas source 180 may bepositioned outside of the enclosure 170 and configured to introduce gasinto the enclosure 170. The gas source 180 may be configured tointroduce a gas that flows (e.g., downward) around the ejector 110, thenozzle 114, the heating elements 130, or a combination thereof. The gasmay flow around and/or within the coils 134. The gas may flow into theenclosure 170 and/or proximate to (e.g., around) the drops 124, the NNS126, and/or the substrate 140.

The 3D printing system 100 may also include a gas sensor 182. The gassensor 182 may be positioned within the enclosure 170. The gas sensor182 may also or instead be positioned proximate to the drops 124, theNNS 126, and/or the substrate 140 (e.g., in an embodiment where theenclosure 170 is omitted). The gas sensor 182 may be configured tomeasure a concentration of the gas, oxygen, or a combination thereof.

The 3D printing system 100 may also include a computing system 190. Thecomputing system 190 may be configured to control the introduction ofthe printing material 120 into the ejector 110, the heating elements130, the power source 132, the substrate control motor 144, thesubtractive tool assembly 160, the fixturing device 166, the gas source180, the gas sensor 182, or a combination thereof. For example, thecomputing system 190 may be configured to automate generation of adesign (e.g., of the NNS 126) such that the design, when subsequentlymanufactured via AM, contains an accessible support volume that may beremoved via SM to yield the 3D part.

Automated Design Generation for Additive Manufacturing with anAccessible Support Volume

The systems and methods described herein may provide automated designgeneration for additive manufacturing with an accessible support volume.This provides a systematic approach to automated design generation whilealso providing removability of the support volume through subtractivemanufacturing (e.g., milling) in terms of accessibility of one or morepoints of the support volume given the substrate 140, the subtractivetool assemblies 160, and/or the fixturing devices 166 without imposingartificial constraints on geometric complexity of the 3D part, thesupport geometry, the subtractive tool assemblies 160, the fixturingdevices 166, or a combination thereof. This also provides efficient andeffective design space exploration by providing complex designs forwhich its NNS 126 (e.g., 3D part+support volume) can be fabricated usingadditive manufacturing and post-processed using subtractivemanufacturing.

Different automated design techniques (e.g., topology optimization,machine learning, cellular automata, etc.) may consider the physicalperformance of a 3D part to provide organic shapes. The presentdisclosure solves the following optimization problem:

$\begin{matrix}{{\underset{\Omega \subseteq \Omega_{0}}{Minimize}{\varphi(\Omega)}},} & \left( {1a} \right)\end{matrix}$ $\begin{matrix}{{{{such}{{{that}\left\lbrack K_{\Omega} \right\rbrack}\left\lbrack u_{\Omega} \right\rbrack}} = \lbrack f\rbrack},} & \left( {1b} \right)\end{matrix}$ $\begin{matrix}{{V_{\Omega} \leq V_{target}},} & \left( {1c} \right)\end{matrix}$ $\begin{matrix} & \left( {1d} \right)\end{matrix}$

Where φ(Ω)∈

is the value of an objective function for a given design Ω⊆Ω₀. Thevariables [f], [u_(Ω)], and [K_(Ω)] represent discretized externalforce, a displacement vector, and a stiffness matrix, respectively, forfinite element analysis (FEA). The variable V_(Ω):=vol[Ω] represents thedesign volume, and V_(target)>0 is the volume budget.

The present disclosure may provide a physics-based performance analysisby invoking physics solvers such as finite element analysis (FEA) toevaluate objectives and constraints. The present disclosure may alsodetermine optimization decision variables such as gradients, sensitivityfields, etc. based at least partially upon the objectives and/orconstraints. The present disclosure may also design manufacturingconstraints (e.g., by augmenting/filtering decision variables based atleast partially upon design and manufacturing considerations). Thepresent disclosure may also update design variables based at leastpartially upon decision variables, and then generate an optimized designof the NSS 126 and/or 3D part.

The accessibility constraint may be augmented to the sensitivity fieldto allow the 3D part to be manufactured using SM. However, the automateddesign of AM parts with respect to accessibility of the support volumehas not yet been explored.

Inaccessibility Access

The 3D printing system 100 (e.g., the nozzle 114, the substrate 140, thesubtractive tool assembly 160, or a combination thereof) can operatewith up to six degrees of freedom (e.g., three translations and threerotations) available for a rigid body. For example, T=(H∪K), where Trepresents the subtractive tool assembly 160 for multi-axis machining, Hrepresents the holder 162, and K represents the cutter 164.Mathematically, the configuration space (C-space) of rigid motions isrepresented as C=

³×SO(3) where C represents the configuration space,

³ represents the Euclidian space, and SO(3) refers to the group of 3×3orthogonal transformations that represent spatial rotations.

An inaccessibility measure field (IMF) may be defined over the 3D designdomain f_(IMF):

³→

for each given tool assembly T as the pointwise minimum of shiftedconvolutions for different choices of sharp points and availableorientations Θ_(t)⊆SO(3). The IMF is described by the followingequation:

$\begin{matrix}{{f_{IMF}\left( {{x;O},N,T,K} \right)}:={\min\limits_{R \in \Theta_{T}}\min\limits_{k \in K}{{vol}\left\lbrack {O\bigcap{\left( {R,x} \right)\left( {T - k} \right)}} \right\rbrack}}} & (2)\end{matrix}$

where R∈Θ is a rotation matrix corresponding to an available toolorientation, and point x∈

³ with N=Ω∪S denoting the near-net shape fabricated of the workpiece andits corresponding support structures S. There are two independenttransformations in effect. First, the shift T→(T−k) in Equation 2 maysample different ways to register the translation space with the designdomain, by changing the local coordinate system to bring different sharppoints to the origin. Second, the rotation (T−k)→(RT−Rk) followed bytranslation (RT−Rk)→(RT−Rk)+x may bring the candidate sharp point (neworigin) to the query point x∈Ω₀.

The same effect can be obtained by querying the convolution at t=(x−Rk)so that the rigid transformation (R, t) brings the sharp point incontact with the query point: (R, t)k=Rk+t=Rk+(x−Rk)=x, as expected. TheIMF may thus be computed as follows:

$\begin{matrix}{{f_{IMF}\left( {{x;O},N,T,K} \right)} = {\min\limits_{R \in \Theta_{T}}{\min\limits_{k \in K}\left( {1_{O}*{\overset{\sim}{1}}_{RT}} \right)}{\left( {x - {Rk}} \right).}}} & (3)\end{matrix}$

Equation 2 can be further extended to consider multiple subtractive toolassemblies 160. Given n_(T)≥1, available subtractive tool assemblies160, T_(i)=(H_(i)∪K_(i)) for 1≤i≤n_(T), their combined IMF may bedetermined by applying another minimum operation over differentsubtractive tool assemblies to identify the subtractive tool assemblieswith the smallest volumetric interference at available orientations andsharp points:

$\begin{matrix}{{f_{IMF}\left( {{x;N},O} \right)}:={\min\limits_{1 \leq i \leq n_{T}}{f_{IMF}\left( {{x;N},O,T_{i},K_{i}} \right)}}} & (4)\end{matrix}$

in which f_(IMF)(x; N, O, T_(i), K_(i)) may be determined using Equation3.

There may be challenges in optimizing the build orientation based atleast partially upon the accessibility of the support volume. As usedherein, the term “build orientation” (also referred to as “build angle”)refers to a direction at which the NNS 126 is additively printedlayer-by-layer. First, minimizing the volume of the support volume maynot be the same as minimizing the volume of inaccessible support volume(i.e., the inaccessible support volume can decrease at a higher overallsupport volume). Second, a large number of build orientations can becometime and computationally intensive. Third, there are numerous types ofsupport geometries (e.g., beams or tree-like structures) and overhangangles (e.g., 45° or 90°) depending on the AM process. Fourth, theshapes of the subtractive tool assembly 160 may not be ignored. Hence,it may be difficult to assign a correspondence between the translationst∈

³ and the points x∈

³ within the near-net shape 126 unless one or more (e.g., all) possiblecontact configurations are analyzed, and the boundary points are treateddifferently from the interior points. Fifth, the shapes of the substrate140, the subtractive tool assembly 160, and the fixture 166 may not beignored. Sixth, the analysis may be highly non-linear, meaning a smallchange in x∈N can dramatically change the accessibility in a far-awaypoint y∈

³.

FIG. 2 depicts a schematic side view of an initial design and boundarycondition 200, according to an embodiment. The “initial design” and/orthe “initial design domain” refers to an envelope within which theoptimized design lies. The “boundary condition” refers to one or morephysical forces and restraints applied to the design. As shown in FIG. 2, the initial design and boundary condition 200 may have one or morefixed points (one is shown: 210). The fixed point 210 may secure theinitial design and boundary condition 200 in a predetermined buildorientation (e.g., 0°) during the AM process. The initial design andboundary condition 200 may also experience one or more forces (five areshown: 220A-220E). The forces may be vertical (e.g., downward) forces.

FIG. 3 depicts a schematic side view of the initial design and boundarycondition 200 in a different build orientation (e.g., 45°), according toan embodiment. The build direction 300 is vertical (e.g., upward) fromthe substrate 140. Referring to FIGS. 2 and 3 , the present disclosuremay be able to determine the stiffest design at V_(target)=0.5. Thebuild direction 300 and substrate 140 may be as shown in FIG. 3 .

FIG. 4 depicts a schematic side view of the NNS 126, according to anembodiment. FIG. 5 depicts a schematic view of the subtractive toolassembly 160, according to an embodiment. The NNS 126 includes a 3D part400 and a support volume (also referred to as support structures). Thesupport volume may include a first support volume 410 and a secondsupport volume 420. The first support volume 410 may be accessible bythe subtractive tool assembly 160, and thus may be removed. The secondsupport volume 420 may be inaccessible by the subtractive tool assembly160, and thus may not be removed. The design in FIG. 4 is at a 0.5volume fraction without considering support volume accessibility.

FIGS. 4 and 5 illustrate a predetermined (e.g., optimized) design Ω ofthe NSS 126 and/or 3D part 400, the accessible support volume 410, theinaccessible support volume 420, and the subtractive tool assembly 160.The set of approach directions are Θ=0°, 30°, 60°, 90°, 120°, 150°,180°, 210°, 240°, 270°, 300°, and 330°. A fraction of the support volumemay be inaccessible, and thus cannot be removed from the NNS 126. As aresult, the 3D part 400 may not be manufacturable.

FIG. 6 depicts an image of the IMF over the NNS 126 in FIG. 4 , and FIG.7 depicts an image of the IMF over the inaccessible support volume 420in FIG. 4 , according to an embodiment. The IMF images may be generatedwithout considering the accessibility of the support volume 410, 420.

Accessibility Constraint on Support Volume Based on IMF

The system and method disclosed herein may automatically generate thedesign of the NNS 126 such that the resulting shape of the NNS 126and/or 3D part 400 may be manufacturable using AM and then SM. Thepresent disclosure may use a physics-based performance analysis thatinvokes physics solvers (e.g., FEA) to evaluate objectives andconstraints. The present disclosure may also determine decisionvariables (e.g., gradients, sensitivity fields, etc.) based at leastpartially upon objectives and/or constraints. The present disclosure mayalso perform an accessibility analysis by constructing the configurationspace (C-space) of the 3D part 400 and subtractive tool assembly 160,sampling the tool rotations in C-space, and constructing the IMFfield(s). The present disclosure may also enrich decision variables withaccessibility information by filtering/augmenting the decision variableswith/by the IMF field(s) such that only the accessible parts are subjectto modification. The present disclosure may also update design variablesbased on the modified decision variables. A SM design may then begenerated.

These activities may be based at least partially upon geometric,topological, material, and/or physical aspects of the availablemanufacturing capabilities. These activities may not be performed inisolation. For example, density-based TO may involve a continuousdensity function ρ_(Ω: Ω→[)0,1] to represent intermediate designs,rather than indicator functions. A threshold 0<τ<1 (e.g., τ=0.5) may beused to define the indicator functions as 1_(Ω)(x)=1 iff ρ_(Ω)(x)>τ foruse in equation 3. However, in other embodiments, direct use of thedensity function may provide additional smoothing:

$\begin{matrix}{{f_{IMF}\left( {{x;\rho_{O}},T,K} \right)}:={\min\limits_{R \in \Theta}{\min\limits_{k \in K}\left( {\rho_{O}*{\overset{\sim}{1}}_{RT}} \right)}{\left( {x - {Rk}} \right).}}} & (5)\end{matrix}$

The function ρ₀: 0→[0,1] can be obtained as ρ₀(x):=ρ_(Ω)(x)+1_(F)(x), inwhich ρ_(Ω)(x) may be obtained directly from TO. The combined IMF forthe tool assemblies f_(IMF)(x; σ₀) f_(IMF)(x; σ_(O)) may be computed as:

$\begin{matrix}{{f_{IMF}\left( {x;\rho_{O}} \right)}:={\min\limits_{1 \leq i \leq n_{T}}{f_{IMF}\left( {{x;\rho_{O}},T_{i},K_{i}} \right)}}} & (6)\end{matrix}$ $\begin{matrix}{{\underset{\Omega \subseteq \Omega_{0}}{Minimize}{\varphi(\Omega)}},} & \left( {7a} \right)\end{matrix}$ $\begin{matrix}{{{{such}{{{that}\left\lbrack K_{\Omega} \right\rbrack}\left\lbrack u_{\Omega} \right\rbrack}} = \lbrack f\rbrack},} & \left( {7b} \right)\end{matrix}$ $\begin{matrix}{{V_{\Omega} \leq V_{target}},} & \left( {7c} \right)\end{matrix}$ $\begin{matrix}{{{V - S_{sec}} = 0},} & \left( {7d} \right)\end{matrix}$

where Vs_(sec) is the volume of the inaccessible support volume 420. Toincorporate the accessibility constraints for multi-axis machining, thesensitivity field S_(Ω) may be modified as follows.

S _(Ω):=(1−w _(acc))S _(φ) _(acc) S _(IMF),   (8)

where 0≤w_(acc)<1 is the filtering weight for accessibility, and can beeither a constant or adaptively updated base on the secluded volumeV_(r(o)). The variable

_(φ) represents the normalized sensitivity field with respect to theobjective function. The volume constraint may be satisfied with theoptimality criteria iteration. The variable

_(IMF) represents the normalized accessibility field defined in terms ofthe normalized IMF as:

$\begin{matrix}{{{\overset{\_}{S}}_{IMF}(x)}:=\left\{ {\begin{matrix}{0.01{{\overset{\_}{f}}_{IMF}\left( {x;\rho_{O}} \right)}} & {{{{if}x} \in {\Omega\bigcup S_{acc}}},} \\{{\overset{\_}{f}}_{IMF}\left( {x;\rho_{O}} \right)} & {{{{if}x} \in S_{sec}},} \\0 & {otherwise}\end{matrix}.} \right.} & (9)\end{matrix}$

FIG. 8 depicts another schematic side view of the NNS 126, according toan embodiment. The design in FIG. 8 is at a 0.5 volume fraction andconsiders support volume accessibility. FIG. 9 depicts an image of theIMF over the NNS 126 in FIG. 8 , according to an embodiment.

FIG. 10 depicts a flowchart of a method 1000 for generating a design,according to an embodiment. The design may be of the NNS 126, the 3Dpart 400, or both. The NNS 126 may then be manufactured via AM, and theNNS 126 may include an accessible support volume 410 that maysubsequently be removed via SM to yield the 3D part 400.

An illustrative order of the method 1000 is provided below; however, oneor more steps of the method 1000 may be performed in a different order,combined, split into sub-steps, repeated, or omitted without departingfrom the scope of the disclosure. One or more steps of the method 1000may be performed using the computing system 190.

The method 1000 may include receiving a representation of the initialdesign domain 200, as at 1002. The representation may be received by(and/or provided to) the computing system 190. The representation mayalso or instead include a build orientation for building the initialdesign domain 200 using the additive manufacturing process. Therepresentation may also or instead include the subtractive tool assembly160.

The method 1000 may also include generating intermediate part designswithin the initial design domain 200, as at 1004. The intermediate partdesigns may be generated by the computing system 190. An intermediatepart design refers to a design of the near-net shape 126 and/or the 3Dpart 400 that is generated after the initial design domain 200 andbefore the final part design (described below). The intermediate partdesigns may be generated iteratively by redistributing material withinthe initial design domain. Each intermediate part design may include thenear-net shape 126, the 3D part 400, the support volume 410, 420, or acombination thereof. In at least one embodiment, the near-net shape 126,the 3D part 400, the support volume 410, 420, or a combination thereofmay vary (e.g., slightly) from one intermediate part design to thesubsequent intermediate part design.

The method 1000 may also include calculating a measure ofinaccessibility of the support volume 410, 420 of each intermediate partdesign, as at 1006. The measure of inaccessibility may be calculated bythe computing system 190. The measure of inaccessibility refers to aninability of the subtractive cutting assembly 160 to access and/orremove one or more regions of the support volume 410, 420 (e.g., in theinitial design domain). The measure of inaccessibility may bequantitative. For example, the measure of inaccessibility may be interms of a volumetric amount of the support volume 410, 420 that isinaccessible, a percentage of the support volume 410, 420 that isinaccessible, a ratio of the support volume 410, 420 that isinaccessible, or the like.

As mentioned above, at least a portion of the method 1000 may beiterative. Thus, at least one (e.g., each) of the intermediate partdesigns may be generated based at least partially upon the measure ofinaccessibility of a previous one of the intermediate part designs. Thisiterative process may continue until termination criteria are met. Inone embodiment, the termination criteria may include the measure ofinaccessibility dropping below a predetermined threshold. This may helpto minimize the amount, percentage, and/or ratio of the inaccessiblesupport volume 420 (e.g., with respect to the accessible support volume410).

The method 1000 may also include generating a final part design, as at1010. The final part design may be generated by the computing system190. The final part design may be generated within the initial designdomain 200. The final part design may be generated based at leastpartially upon the intermediate part design(s) and/or the measure(s) ofinaccessibility.

The method 1000 may also include building the final part design with the3D printing system 100, as at 1012. The method 1000 may also includeremoving the support volume 410, 420 from the final part design with the3D printing system 100, as at 1014. More particularly, this may includeremoving the accessible support volume 410 from the final part designusing the subtractive tool assembly 160. This may yield the 3D part 400.

Path Planning

In one embodiment, the system and method may invoke path planners suchas open motion planning library (OMPL) to test for sufficientaccessibility conditions. The new field, coupled with IMF, can be usedto generate designs that satisfy both conditions of existence of aconnected path.

In one embodiment, the system and method may provide an automatedapproach to removing the support volume 410, 420 where the IMF iscomputed over one or more (e.g., all) of the support volume 410, 420,and the conditions for existence of a plan is evaluated. In anotherembodiment, the system and method may automatically optimize a shape tomeet multiple physical performance criteria while ensuring that theresulting shape has an accessible support volume 410 given the multiplesubtractive tool assemblies 160, tool orientations, and builddirection(s). In another embodiment, the system and method may providean automated approach to support volume removal planning where the IMFis computed over one or more (e.g., all) of the support volume 410, 420,and the support volume 410, 420 is removed based upon a predetermined(e.g., optimized) path. For example, the accessible support volume 410may be removed using path planners such as OMPL. In another embodiment,the system and method may automatically generate a shape to meetmultiple physical performance criteria while ensuring that the resultingsupport volume 410, 420 is removable from the NNS 126 with the given setof subtractive tool assemblies 160, orientations, and fixturing devices166.

Physics-Aware Automatic Spatial Planning for Subtractive and HybridManufacturing

As mentioned above, an additive manufacturing (AM) process may produce anear-net shape 126, defined as a shape that closely conforms to theintended design to be manufactured (e.g., the 3D part 400) along withadditional support volume (also referred to as support structures and/orscaffolding) 410, 420 added during the AM process. The support volume410, 420 may be subsequently removed so that the intended design (e.g.,3D part 400) may be fabricated from the NNS 126. The systems and methodsdisclosed herein may provide a spatial planning approach toautomatically remove the support volume 410, 420 using the multi-axissubtractive tool assembly 160 while ensuring that 1) the work-piece(e.g., NNS 126) is held by the fixture 166 throughout the supportremoval process (i.e., it is not detached from the platform before someor all supports 410, 420 are removed) and 2) the part 400 is not damagedin the removal process. The multi-disciplinary approach involves solvingthe corresponding physics problems, augmenting additional physics-basedsensitivity fields such as topological sensitivity field (TSF) to theinaccessibility measure field (IMF), and producing an efficient supportremoval plan.

Producing the 3D part 400 using AM may include some post-processingoperations, typically in the form of machining or other subtractivemanufacturing (SM) processes. AM may therefore be understood as one ofmany processes that may be used to manufacture a part and not astand-alone solution. A sequence of AM and SM processes (in noparticular order) is defined as a hybrid manufacturing process. Hybridmachines may couple a LENS (Laser Engineered Net Shaping) AM processwith a high-axis milling center to enable AM on curved surfaces.

In a hybrid manufacturing process, the interaction between SM and AM maybe analyzed, for example, when planning the layout and removal ofsupporting/scaffolding material 410, 420 generated by the AM process tocreate the NNS 126. The AM process may generate the support volume 410,420 to sustain the manufactured part (e.g., NNS 126) so that it does notcollapse under its own weight as material is added during the AMprocess. The resulting NNS 126 (e.g., the 3D part 400 along with thesupport volume 410, 420) may then be manually post-processed to removethe support volume 410, 420 and then finish the 3D part 400. It ispossible that with some AM process plans, the support volume 410, 420may be placed at locations that are inaccessible to the subtractive toolassemblies 160 used in the SM process. Furthermore, the geometry of thepart 400, the support volume 410, 420, and the subtractive tool assembly160 may create a complex space of feasible (e.g., non-colliding) toolconfigurations (e.g., positions and orientations) that determine supportvolume removability. Therefore, the problem of removing AM supportvolume 410, 420 in a SM process is a spatial planning problem involvingthe analysis of the tool's feasible configurations against a dynamic NNS126 that is updated whenever a support volume 410, 420 is removed. Thefollowing description focuses on SM operations that remove previouslygenerated AM support volume 410, 420.

Inaccessibility Measure Field

The accessibility analysis for imposing support volume accessibilityconstraints through multi-axis machining is provided below. For thesubtractive tool assembly 160, T=(H∪K) can operate with up to sixdegrees of freedom (e.g., three translations and three rotations)available for a rigid body, where H and K represent the holder 162 andthe cutter 164, respectively, Ω represents the 3D part 400, F representsthe substrate 140 (and other fixtures), and S represents the supportvolume 410, 420.

Mathematically, the configuration space (C-space) of rigid motions maybe represented as C=

³×SO(3); SO(3) refers to the group of 3×3 orthogonal transformationsthat represent spatial rotations. The inaccessibility measure field(IMF) may be defined over the 3D design domain f_(IMF):

³→

for each given tool assembly T and orientation R∈Θ, where availableorientations for the tool T is Θ⊂SO 3, as the pointwise minimum ofshifted convolutions for different choices of sharp points (whichdepends on T):

$\begin{matrix}{{f_{IMF}\left( {{x;O},T,K,R} \right)}:={\min\limits_{k \in K}{{{vol}\left\lbrack {O\bigcap{\left( {R,x} \right)\left( {T - k} \right)}} \right\rbrack}.}}} & (10)\end{matrix}$

where point x∈O, and obstacle O=Ω∪F. There are two independenttransformations in effect. First, the shift T→(T−k) in Equation 10 maytry different ways to register the translation space with the designdomain, by changing the local coordinate system to bring different sharppoints to the origin. Second, the rotation (T−k)→(RT−Rk) followed bytranslation (RT−Rk)→(RT−Rk)+x may bring the candidate sharp point (neworigin) to the query point x∈Ω.

The same effect can be obtained by querying the convolution at t=(x−Rk)so that the rigid transformation (R, t) brings the sharp point incontact with the query point: (R, t)k=Rk+t=Rk+(x−Rk)=x, as expected. TheIMF may thus be computed as follows:

$\begin{matrix}{{f_{IMF}\left( {{x;O},T,K,R} \right)} = {{\min\limits_{k \in K}\left( {1_{O}*{\overset{\sim}{1}}_{RT}} \right)}{\left( {x - {Rk}} \right).}}} & (11)\end{matrix}$

Physics-Based Sensitivity Field

For a given physical quantity of interest φ, a topological sensitivityfield (TSF) defined at every point x of the design Ω may be determinedto measure the change in φ if an infinitesimally small amount ofmaterial is removed from that point. TSF can be defined as:

$\begin{matrix}{{{{TSF}\left( {x;\Omega} \right)}:={\lim\limits_{\epsilon\longrightarrow 0^{+}}\frac{{\varphi\left( {\Omega - {B_{\epsilon}(x)}} \right)} - {\varphi(\Omega)}}{{vol}\left\lbrack {\Omega\bigcap{B_{\epsilon}(x)}} \right\rbrack}}},} & (12)\end{matrix}$

B_(ϵ)(x)⊂Ω is a small 3D ball of radius ϵ→0⁺ centered at a given querypoint x∈Ω. The numerator of the limit evaluates the (e.g., presumablyinfinitesimal) change in φ(Ω) when the candidate design is modified asΩ→(Ω−B_(ϵ)(x)) (e.g., by puncturing an infinitesimal cavity at the querypoint). The denominator vol[Ω∩B_(ϵ) (x)]=O (ϵ³) as ϵ→0⁺ measures thevolume of the cavity.

TSF may be used in topology optimization where the material is removedfrom regions with lower TSF values. However, TSF has not previously beenused in the context of spatial planning.

Physics-Aware Automatic Spatial Planning

One of the challenges in automated spatial planning for subtractive andhybrid manufacturing is considering the impact of the physical forces.Examples of these forces can be the contact forces of the subtractivetool assembly 160 with the NNS 126 at removal points or gravity when thesacrificial support volume 410, 420 is connecting the 3D part 400 to thesubstrate 140 held by the fixture 166. Given the two fields IMF and TSF,a physics-aware IMF (PIMF) may be defined to find accessible regionswith the least negative impact on the work-piece, and subsequentlygenerate automated manufacturing plans that are feasible and practical.Mathematically, PIMF can be written as:

PIMF(x; Ω):=w ₁IMF(x; Ω)+(1−w ₁)TSF(x; Ω),   (13)

Both IMF and TSF may be normalized, and w₁ may be from about 0.1 toabout 0.9, from about 0.2 to about 0.8, from about 0.3 to about 0.7, orfrom about 0.4 to about 0.6 (e.g., about 0.5).

Physics-Aware Automatic Support Removal Planning for HybridManufacturing

Automatic support volume removal planning is an example of spatialmanufacturing planning, where given a set of tool assemblies 160, toolorientations, and fixturing devices 166, a greedy algorithm may beconstructed to remove the support volume 410, 420 while ensuring thatthe NNS 126 remains attached to the substrate 140 held by the fixturingdevice 166 until the last support volume 410, 420 is removed. In otherwords, a sequence of support removal may be determined by selecting themost efficient tool 160, orientation, and fixture 166 while ensuringthat NNS 126 and/or 3D part 400 does not prematurely detach from thesubstrate 140 and fall under its weight. Subsequently, the TSF may bedetermined, which captures the change in the overall structuralstiffness if material is (e.g., hypothetically) removed from each pointin NNS 126. To find the total accessible regions S_(i) ^(acc) for eachtool T_(i), i=1, . . . , n_(T), orientation R_(ij) (j^(th) orientationof the i^(th) tool), and fixture 166, the IMF with respect tonon-sacrificial obstacle O=Ω∪F may be determined according to Equation11. This is a condition to prevent a collision between the selected tool160 in a particular orientation with the NNS 126 and fixturing devices166. The accessible support volume regions for each tool 160 andorientation can be written as:

S _(i) ^(acc) ⊆S={∀x∈S: f_(IMF)(x; Ω, F, T _(i) , K _(i) , R_(ij))≤σ_(acc)}. (5)   (14)

where τ_(acc) is a small threshold value given the numerical errors fromdiscretization of models. To find the next step in removing thesacrificial support volume 410, 420, the near-net IMFs f_(IMF) (x; Ω, S,F, T_(i), R_(ij)) may be determined for each tool 160 and its availableorientations over the NNS 126 to find removable support volume Si toensure no collision between the tool T_(i) under orientation R_(ij) andthe remaining support volume 410, 420. Subsequently, the TSF may beaugmented with these near-net IMFs, and the following PIMF may bedetermined:

PIMF_(ij) :=w ₁ f _(IMF)(N, F, T _(i) , K _(i) , R _(ij))+(1−w ₁)f_(TSF)(N, F, T _(i) K _(i) , R _(ij)),   (15)

Considering a maximum allowed removal volume and a threshold level-setvalue τ_(rem), the removable support volume for the tool T_(i) may bedetermined under orientation R_(ij):

S _(rem) ^(ij) ⊆S _(acc) ^(i) ={∀x∈S: PIMF _(ij)≤σ_(rem)}.   (16)

FIG. 11 depicts a portion of the 3D printing system 100, according to anembodiment. More particularly, FIG. 11 shows a support volume removalsetup with 3 different subtractive tool assemblies 160A-160C and aplurality of (26) different subtractive tool orientations. In thisexample, the first subtractive tool assembly 160A has 14 differentorientations, the second subtractive tool assembly has 6 differentorientations, and the third subtractive tool 160C has 6 differentorientations. FIG. 11 also shows another example of the 3D part 1210,the substrate (i.e., build platform) 140, the fixturing device 166holding the substrate 140.

FIG. 12 depicts another example of the NNS 1200 with the substrate 140,according to an embodiment. The NSS 1200 includes the 3D part 1210 andthe support volume 1220.

FIGS. 13A-13E depict different fields that are computed for the firstsubtractive tool assembly 160A (T₁) and its first orientation R₁₁,according to an embodiment. More particularly, FIG. 13A shows the IMFwith obstacle O=Ω∪F for T₁ and R₁₁. FIG. 13B shows the IMF with obstacleN=N∪F. FIG. 13C shows the deformation field on the NNS 1200. FIG. 13Dshows the TS field on the support volume 1220. FIG. 13E shows thephysics-aware IMF for T₁ and R₁₁ (PIMF₁₁).

FIGS. 14A-14E depict progressive removal of the support volume 1220 fromthe NSS 1200 to yield the 3D part 1210, according to an embodiment. Inthis particular example, the structure removal plan has/usesτ_(acc)=0.025 and τ_(rem)=0.1. In FIG. 14A, the first subtractive toolassembly 160A has removed 19.78% of the support volume 1220. In FIG.14B, the first subtractive tool assembly 160A has removed an additional31.59% of the support volume 1220. In FIG. 14C, the second subtractivetool assembly 160B has removed an additional 29.57% of the supportvolume 1220. In FIG. 14D, the third subtractive tool assembly 160C hasremoved an additional 13.52% of the support volume 1220. In FIG. 14E, afourth subtractive tool assembly 160D has removed another 2.24% of thesupport volume 1220. As a result, the 3D part 1210 remains. Otherheuristics or quantities of interest such as the time or cost of eachsubtractive tool assembly 160A-160D, reorientation, or different physicscan also be added to the proposed framework to construct more informedmanufacturing plans. Further, motion planning tools such as an OMPL canbe used to increase the accuracy of generated plans.

FIG. 15 depicts a flowchart of a method 1500 for planning for removal ofthe support volume 1220 in hybrid manufacturing, according to anembodiment. An illustrative order of the method 1500 is provided below;however, one or more steps of the method 1500 may be performed in adifferent order, combined, split into sub-steps, repeated, or omittedwithout departing from the scope of the disclosure. One or more steps ofthe method 1000 may be performed using the computing system 190.

The method 1500 may include receiving the near-net shape 1200, as at1502. The representation may be received by (and/or provided to) thecomputing system 190. The representation may also or instead include the3D part 1210 and/or the support volume 1220. The representation may alsoor instead include the subtractive tool assembly 160.

The method 1500 may also include calculating a measure ofinaccessibility of the support volume 1220 by the at least onesubtractive tool assembly 160, as at 1504. The measure ofinaccessibility may be calculated using the computing system 190. Themeasure of inaccessibility is defined above.

The method 1500 may also include calculating a measure of change in aphysical quantity of interest with respect to a change in the near-netshape, as at 1506. The measure of change may be calculated by thecomputing system 190. The physical quantity of interest may be orinclude deformation, strain energy, stress, strain, buckling, thermalconduction, thermal convection, etc. The change may be or include ahypothetical change. The change may be smaller than a predeterminedsize. For example, the change may be infinitesimal.

The method 1500 may also include constructing a physics-awareinaccessibility measure based at least partially upon the measure ofinaccessibility, the measure of change, or both, as at 1508. Thephysics-aware inaccessibility measure may be constructed using thecomputing system 190. The physics-aware inaccessibility measure may beconstructed by combining the measure of inaccessibility and the measureof change. The physics-aware inaccessibility measure indicates aremovability (i.e., an ability to remove) of a region of the supportvolume 1220 from the near-net shape 1200.

The method 1500 may also include creating a plan to remove a region ofthe support volume 1220 based at least partially upon the physics-awareinaccessibility measure, as at 1510. The plan may be created using thecomputing system 190. More particularly, this step may include creatinga plan to remove the region of the support volume 1220 with the at leastone subtractive tool assembly 160.

The method 1500 may also include building the near-net shape 1200 withthe 3D printing system 100, as at 1512. The near-net shape 1200 may bebuilt based at least partially upon the plan.

The method 1500 may also include removing the support volume 1220 fromthe near-net shape 1200 with the 3D printing system 100, as at 1514.More particularly, this may include removing the accessible supportvolume 1220 from the near-net shape 1200 using the subtractive toolassembly 160. This may yield the 3D part 1210.

Notwithstanding that the numerical ranges and parameters setting forththe broad scope of the present teachings are approximations, thenumerical values set forth in the specific examples are reported asprecisely as possible. Any numerical value, however, inherently containscertain errors necessarily resulting from the standard deviation foundin their respective testing measurements. Moreover, all ranges disclosedherein are to be understood to encompass any and all sub-ranges subsumedtherein. For example, a range of “less than 10” may include any and allsub-ranges between (and including) the minimum value of zero and themaximum value of 10, that is, any and all sub-ranges having a minimumvalue of equal to or greater than zero and a maximum value of equal toor less than 10, e.g., 1 to 5.

While the present teachings have been illustrated with respect to one ormore implementations, alterations and/or modifications may be made tothe illustrated examples without departing from the spirit and scope ofthe appended claims. For example, it may be appreciated that while theprocess is described as a series of acts or events, the presentteachings are not limited by the ordering of such acts or events. Someacts may occur in different orders and/or concurrently with other actsor events apart from those described herein. Also, not all processstages may be required to implement a methodology in accordance with oneor more aspects or embodiments of the present teachings. It may beappreciated that structural objects and/or processing stages may beadded, or existing structural objects and/or processing stages may beremoved or modified. Further, one or more of the acts depicted hereinmay be carried out in one or more separate acts and/or phases.Furthermore, to the extent that the terms “including,” “includes,”“having,” “has,” “with,” or variants thereof are used in either thedetailed description and the claims, such terms are intended to beinclusive in a manner similar to the term “comprising.” The term “atleast one of” is used to mean one or more of the listed items may beselected. Further, in the discussion and claims herein, the term “on”used with respect to two materials, one “on” the other, means at leastsome contact between the materials, while “over” means the materials arein proximity, but possibly with one or more additional interveningmaterials such that contact is possible but not required. Neither “on”nor “over” implies any directionality as used herein. The term“conformal” describes a coating material in which angles of theunderlying material are preserved by the conformal material. The term“about” indicates that the value listed may be somewhat altered, as longas the alteration does not result in nonconformance of the process orstructure to the illustrated embodiment. The terms “couple,” “coupled,”“connect,” “connection,” “connected,” “in connection with,” and“connecting” refer to “in direct connection with” or “in connection withvia one or more intermediate elements or members.” Finally, the terms“exemplary” or “illustrative” indicate the description is used as anexample, rather than implying that it is an ideal. Other embodiments ofthe present teachings may be apparent to those skilled in the art fromconsideration of the specification and practice of the disclosureherein. It is intended that the specification and examples be consideredas exemplary only, with a true scope and spirit of the present teachingsbeing indicated by the following claims.

1. A method, comprising: receiving a representation of a near-net shapecomprising a 3D part and a support volume; calculating a measure ofinaccessibility of the support volume by at least one subtractive toolassembly; calculating a measure of change in a physical quantity ofinterest with respect to a change in the near-net shape; constructing aphysics-aware inaccessibility measure based at least partially upon themeasure of inaccessibility and the measure of change; creating a plan toremove at least a portion of the support volume using the at least onesubtractive tool assembly based at least partially upon thephysics-aware inaccessibility measure; building the near-net shape usinga 3D printer; and removing a region of the support volume from thenear-net shape using the at least one subtractive tool assembly based atleast partially upon the plan to yield the 3D part.
 2. The method ofclaim 1, wherein the at least one subtractive tool assembly comprises amulti-axis machining tool.
 3. The method of claim 2, wherein themulti-axis machining tool comprises a 2-axis turning machine, a 3-axismilling machine, a 5-axis turn-milling machine, a sawing machine, awire-cutting machine, a laser-cutting machine, or a combination thereof.4. The method of claim 1, wherein the measure of inaccessibilitycomprises a continuous, spatial field that quantifies to what extentdifferent points in the initial design domain cannot be accessed by theat least one subtractive tool assembly.
 5. The method of claim 4,wherein the at least one subtractive tool assembly comprises a pluralityof subtractive tool assemblies, and wherein the measure ofinaccessibility is calculated for the plurality of subtractive toolassemblies as a minimum of the measure of inaccessibility for each ofthe plurality of subtractive tool assemblies.
 6. The method of claim 5,wherein the measure of inaccessibility for each of the plurality ofsubtractive tool assemblies at every query point of the initial designdomain is calculated as a minimum of the measure of inaccessibility fordifferent configurations at which the query point is removable by theplurality of subtractive tool assemblies.
 7. The method of claim 6,wherein the different configurations comprise: at least one displacementthat brings the at least one subtractive tool assembly in contact withthe query points; and one or more orientations that are available to theat least one subtractive tool assembly.
 8. The method of claim 1,wherein the measure of inaccessibility is at least partially defined bya volume of a collision between the at least one subtractive toolassembly and the 3D part.
 9. The method of claim 1, further comprisingreceiving a representation of at least one fixture for subtractivemanufacturing and a build platform, and wherein the measure ofinaccessibility is at least partially defined by a volume of a collisionbetween the at least one subtractive tool assembly and the near-netshape, the build platform, the at least one fixturing device, or acombination thereof.
 10. The method of claim 1, wherein the measure ofchange in the physical quantity of interest is based at least partiallyupon a topological sensitivity field.
 11. The method of claim 10,wherein the topological sensitivity field is with respect to deformationor stress caused by a weight of the near-net shape.
 12. The method ofclaim 10, wherein the topological sensitivity field is with respect todeformation or stress caused by a contact force of the at least onesubtractive tool assembly.
 13. The method of claim 1, wherein thephysics-aware inaccessibility measure comprises a weighted sum of themeasure of inaccessibility and the measure of change.
 14. A method ofplanning for removal of a support volume in hybrid manufacturing, themethod comprising: providing a computer with a representation of: a 3Dpart; a near-net shape comprising the 3D part and the support volume;and at least one subtractive tool assembly with a number of degrees offreedom; calculating a measure of inaccessibility of the support volumeby the at least one subtractive tool assembly using the computer;calculating a measure of change in a physical quantity of interest withrespect to a change in the near-net shape using the computer;constructing a physics-aware inaccessibility measure by combining themeasure of inaccessibility and the measure of change using the computer,wherein the physics-aware inaccessibility measure indicates aremovability of a region of the support volume from the near-net shape;creating a plan to remove the region of the support volume with the atleast one subtractive tool assembly using the computer, wherein the planis based at least partially upon the physics-aware inaccessibilitymeasure; building the near-net shape using a 3D printer; and removingthe region of the support volume from the near-net shape based at leastpartially upon the plan to yield the 3D part.
 15. The method of claim14, wherein the change comprises a hypothetical change that is less thana predetermined size.
 16. The method of claim 14, wherein the physicalquantity of interest comprises stress.
 17. (canceled)
 18. A method ofplanning for removal of a support volume in hybrid manufacturing wherethe support volume is added during an additive manufacturing process andsubsequently removed through a subtractive manufacturing process, themethod comprising: providing a computer with a representation of: a 3Dpart; a near-net shape comprising the 3D part and the support volume;and at least one subtractive tool assembly with a number of degrees offreedom; calculating a measure of inaccessibility of the support volumeby the at least one subtractive tool assembly using the computer;calculating a measure of change in a physical quantity of interest withrespect to a hypothetical change in the near-net shape, wherein thehypothetical change is less than a predetermined size; constructing aphysics-aware inaccessibility measure by combining the measure ofinaccessibility and the measure of change using the computer, whereinthe physics-aware inaccessibility measure indicates a removability of aregion of the support volume from the near-net shape; creating a plan toremove the region of the support volume with the at least onesubtractive tool assembly using the computer, wherein the plan is basedat least partially upon the physics-aware inaccessibility measure;building the near-net shape via the additive manufacturing process usinga 3D printer; and removing the region of the support volume from thenear-net shape via the subtractive manufacturing process based at leastpartially upon the plan to yield the 3D part.
 19. (canceled)
 20. Themethod of claim 18, wherein the hypothetical change comprises a changethat is less than a predetermined threshold.
 21. The method of claim 1,wherein the physics-aware inaccessibility measure is constructed basedat least partially upon a sum of the measure of inaccessibility and themeasure of change.